DW5E.8.pdf
2016 Imaging and Applied Optics Congress (3D, AO, AIO, COSI, DH, IS, LACSEA, MATH) © OSA 2016
Unwrapping algorithm based on least-squares, iterations, and phase calibration to unwrap phase highly corrupted by decorrelation noise H.-T. Xia1,2,3, R.-X. Guo1,2, S. Montresor4, P. Picart4,5, J.-C. Li3, F. Yan1,2, H.-M. Cheng1 1-Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming 650500, China 2-Key Laboratory of Yunnan Higher Education Institutes for Mechanical Behavior and Microstructure Design of Advanced Materials, Kunming 65050, China 3-KUST, Kunming University of Science and Technology, Kunming 65050, China 4-Université du Maine, CNRS UMR 6613, LAUM, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France 5-École Nationale Supérieure d’Ingénieurs du Mans, rue Aristote, 72085 Le Mans Cedex 9, France
[email protected],
[email protected]
Abstract: A phase unwrapping algorithm based on least-squares, iterations and phase calibration is proposed. We demonstrate that the proposed method is able to process phase data including high amount of decorrelation phase noise. The algorithm is validated by numerical simulations, and comparison to other algorithms shows that the proposed algorithm has better accuracy. The proposed algorithm is applied in digital holography and the unwrapped results demonstrate its practical effectiveness. OCIS codes: (090.1995) Digital holography, (090.2880) Holographic interferometry, (100.5088) Phase unwrapping, (100.0100) Image processing, (100.3010) Image reconstruction techniques
1. Introduction Digital holography methods are very efficient techniques for the measurement of deformation fields of the object surface and for the measurement of surface shapes and contours [1]. As a general rule, the object field is numerically reconstructed using a propagation operator, for example the discrete Fresnel transform, or the angular spectrum transfer function [1]. Quantitative measurement is carried out by a subtraction between the object phase and a “reference” phase. The phase difference is then obtained modulo 2π and requires phase unwrapping [2]. In the case of digital holographic microscopy the reference phase may be the phase extracted without any object in the measurement beam. For the case of an object having a rough surface, the reference is that of the object at the “reference” state. When dealing with dynamic measurements, the subtraction is performed between phases calculated at each instant of the hologram sequence [1,3], the phase produced at each instant by the object being time-varying. However, a limitation of such an approach is related to the speckle decorrelation occurring when the object is time-varying. This decorrelation adds a high spatial frequency noise to the useful signal. The raw phase difference is then not directly suitable for visualization or comparison with some theoretical/simulation results. Smoothing methods were developed in the past [4], but when the fringes are close together there exists a risk of generating phase dislocations. Thus, the unwrapping process requires being highly robust to decorrelation noise. So, the phase unwrapping algorithm which directly operates on the phase map highly corrupted by noise and holes has to be developed. In previous work [5], we presented a phase unwrapping method based on least-squares and iterations of unwrapped phase errors. By this method, phase unwrapping can be carried out in the presence of noise and holes and yields accurate unwrapped phase maps. However, in subsequent applications, we found that this method would fail for low-SNR phase maps. Unfortunately, such degraded conditions are often encountered in high speed digital holography [3]. In this paper, a robust noise immune unwrapping algorithm based on calibration of phase difference is proposed. This method is validated by simulated and experimental phase maps corrupted by noise and holes. 2. Basics of the method The least-squares phase unwrapping method seeks the unwrapped solution by minimizing the differences between the discrete partial derivatives of the wrapped phase data and those of the unwrapped solution [5]. Let’s note ϕij to represent the true phase and ψij to represent the wrapped phase at the grid point (i,j) of a phase map. The wrapping operator is defined as W(φij ) = ψ ij = φij + 2πkij , where kij is an integer, − π < ψ ij ≤ π , M, N are respectively the number of grid points with respect to the i index and the j index. The wrapped phase differences are defined as:
DW5E.8.pdf
∆xij = W {ψ (i +1) j − ψ ij }
2016 Imaging and Applied Optics Congress (3D, AO, AIO, COSI, DH, IS, LACSEA, MATH) © OSA 2016
(i = 0,1,2, L , M − 2;
∆ =0 x ij
j = 0,1,2, L , N − 1)
otherwise
∆ = W {ψ i ( j +1) − ψ ij }
(1)
(i = 0,1,2, L , M − 1;
y ij
∆ =0 y ij
j = 0,1,2, L , N − 2)
otherwise,
where ∆ijx and ∆ijy are respectively the difference with respect to i and j indexes. In the least-squares sense, the optimal solution ϕij can be obtained from the discrete Poisson equation with Neumann boundary conditions. The discrete Poisson equation is solved by the DCT method [5]. In this paper, we propose to calibrate the large gradient differences to the appropriate values. The calibration to the gradient phase differences given by Eq. (1) can be defined as:
∆xij = Gijx
(if
∆xij ≥ T )
∆xij = ∆xij
otherwise
y ij
∆ =G
(if
y ij
∆ =∆
otherwise
y ij
y ij
(2)
∆ ≥ T) y ij
where T is a threshold, and Gijx and Gijy are the calibrated values. T can be determined by the standard deviation σ of the noise, that is T=σ−π. Usually we chose to set Gijx = Gij y=0.1T. The estimation of σ will be not discussed in this paper. In the following, this approach is referred as MPULSI (Modified Phase Unwrapping based on LeastSquares and Iterations).
3. Realistic simulation of decorrelation noise In order to evaluate the performances of the unwrapping algorithm, a realistic numerical simulation was developed. The goal of the simulation is to produce phase map corrupted by speckle decorrelation noise with adequate probability density function and average speckle size. In experiments, the amount of speckle phase decorrelation is naturally controlled by the fringe density, which is related to the surface modifications. The largest the surface deformation, the highest the number of fringes, and the highest the phase decorrelation between the two states of the object surface. In this paper, we aim at simulating phase decorrelation effects that corrupt the digital modulo 2π fringes. To do this, the surface of the object is supposed to be rough compared to the wavelength of light and is illuminated by a uniform plane wave. In order to get phase change due to surface deformation and including speckle phase decorrelation, the procedure is as follows: first calculate the convolution equation with the random surface to produce a random “reference” speckle field in the image plane, second calculate the convolution equation with the random surface to which is added the surface deformation to produce a random “deformed” speckle field in the image plane, last calculate the phase difference between the two previous speckle phases of the two calculated optical fields. The simulation provides useful exact data to estimate the phase error in the unwrapping process.
4. Evaluation of the proposed approach We use phase map modeled with the Matlab peaks function to generate phase map and corresponding corrupted phase with noise standard deviation at σ ≈1rad. In order to compare the robustness of the proposed algorithm, three other different phase unwrapping algorithms were selected. These are: Goldstein branch cut algorithm (namely Goldstein), quality guided path following algorithm (Quality), phase unwrapping based on least-squares and iteration (PULSI) [2,5]. The effectiveness of the MPULSI algorithm for noisy phase maps can be appreciated in Fig. 1. Fig 1(a) shows the simulated noisy unwrapped phase using the Matlab peaks function and Fig. 1(b) exhibits the modulo 2π corresponding phase. High noise level can be observed (σ ≅ 1rad). Fig. 1(c), Fig. 1(d), Fig. 1(e), and Fig. 1(f) show respectively the unwrapped phases obtained from Goldstein, Quality, PULSI and MPULSI algorithms. As can be seen in Fig. 1, the only correct unwrapped phase with such noise level is obtained from MPULSI algorithm. From the exact simulated phase, the phase error due to unwrapping can be estimated. Fig. 2(a), Fig. 2(c), Fig. 2(c), and Fig. 2(d) exhibit the phase errors from the four evaluated algorithms. Fig. 2 shows that the phase error given by MPUSI is only related to noise whereas the three other phase errors include systematic errors. Especially, there exist regional larger errors in the unwrapped phase by Goldstein, Quality and PULSI, but only sparsely pointlike errors in that from MPULSI. Such noise-like error can be reduced with low pass filtering applied to unwrapped
DW5E.8.pdf
2016 Imaging and Applied Optics Congress (3D, AO, AIO, COSI, DH, IS, LACSEA, MATH) © OSA 2016
phase. So, these simulation results demonstrate that the MPULSI algorithm is able to unwrap noisy phase, when others fail.
Fig. 1 Simulated phase map with realistic noise (σ≈1.0rad), (a) original unwrapped noisy phase, (b) wrapped noisy phase, (c) unwrapped phase with Goldstein algorithm, (d) unwrapped phase with Quality algorithm, (e) unwrapped phase with PULSI, (f) unwrapped phase with MPULSI.
The four algorithms were applied to experimental data from digital Fresnel holography. The data include a large amount of decorrelation phase noise. Fig. 2(e), Fig. 2(f), Fig. 2(g), and Fig. 2(h) exhibit the unwrapped phase from the four algorithms (resp. Goldstein, Quality, PULSI, MPULSI). It can be seen that the unwrapped results given by MPULSI is better than those from the three other algorithms.
Fig. 2. Error in the unwrapped phase from different algorithms, (a) Goldstein, (b) Quality, (c) PULSI, (d) MPULSI); Unwrapped experimental phase with, (e) Goldstein, (f) Quality, (g) PULSI, (h) MPULSI.
5. Conclusion This paper proposes a robust phase unwrapping algorithm able to unwrap data with high amount of decorrelation phase noise from digital holographic experiments. This work was supported by the Natural Science Foundation of China (11362007, 11462009). 6. References [1] P. Picart, New techniques in digital holography, (ISTE-Wiley, London, 2015). [2] D.C. Ghiglia, and M.D. Pritt, Two-dimensional phase unwrapping: theory, algorithms and software, (Wiley, New York, 1998). [3] J. Poittevin, P. Picart, F. Gautier, and C. Pézerat, "Quality assessment of combined quantization-shot-noise-induced decorrelation noise in high-speed digital holographic metrology," Opt. Expr. 23(24), 30917-30932 (2015). [4] H.A. Aebischer, and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Comm. 162, 205-210 (1999). [5] H.-T. Xia, R.-X. Guo, Z.-B. Fan, and B.-C. Yang, "Non-invasive mechanical measurement for transparent objects by digital holographic interferometry based on iterative least-squares phase unwrapping," Exp. Mech. 52(4), 439-445 (2012).
